# Basic Principles of Numbers

These concepts form the foundation of more complicated problems, so it is important to know. For example, you could end up with a completely wrong answer if you solve for real numbers when the question asks for integers.

### Types of Numbers

For the GMAT, you need to know the more common types of numbers, such as integers, rational numbers, real numbers, and prime numbers. Also, you should at least be aware of some of the less common types, such as irrational and imaginary numbers.

1. Integers and Whole Numbers

Both integers and whole numbers can’t have any decimals or fractions attached, They are the numbers you would use to describe things that aren’t normally split into parts, like marbles, cars and people. The only difference is that integers can be negative, while whole numbers are only positive or zero.

Integers are numbers that belong to the set of all positive and negative whole numbers with 0 included. Integers include –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, and 5 and continue infinitely on either side of 0. Integers greater than 0 are called natural numbers or positive integers. Integers less than 0 are called negative integers.

2. Rational and Irrational Numbers

Rational numbers are any numbers that can be expressed as a fraction or ratio of one integer to another, that is, numbers that can be expressed as fractions. Rational numbers include all positive and negative integers, zero, fractions, and decimal numbers that either end or repeat.

Irrational numbers extend out for many places behind the decimal, such as π or √2. The decimal equivalents of these numbers don’t end or repeat.

3. Real and Imaginary Numbers

Real numbers belong to the set that includes all integers, rational numbers, and irrational numbers. Real numbers as those numbers that can be represented by all the points on a number line, either positive or negative or zero.

Imaginary number is any number that isn’t a real number. For example, when you square any positive or negative real number, the result is a positive number. This means you can’t find the square root of a negative number unless the root is simply not a real number. So imaginary numbers include square roots of negative numbers or any number containing i, which represents the square root of –1.

4. Prime Numbers and Composite Numbers

All the positive integers that can be divided by only themselves and 1 are prime numbers. So, a prime number has exactly two distinct factors: itself and 1. 1 is not a prime number. The smallest prime number is 2, and it’s also the only even prime number. The set of prime numbers includes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and so on.

Numbers that are not prime numbers are called composite numbers. A composite number has more than two factors, so it’s the product of more than simply itself and the number 1.

### Basic Operations

Now that you are more comfortable with some terms, it is time for manipulating numbers. You are probably familiar with the standard operations of addition, subtraction, multiplication, and division. But even these basics have some tricky elements that you may need to refresh your memory on.

Addition is just the operation of combining two or more numbers to get an end result called the sum. For example, 3 + 4 + 5 = 12.

Addition has two important properties the associative property and the commutative property.

• Associative property: The associative property states that the order in which you choose to add three or more numbers doesn’t change the result. It shows how numbers can group differently with one another and still produce the same answer. So regardless of whether you add 3 and 4 together first and then add 5 or add 4 and 5 together followed by 3, you still get an answer of 12. So, (3 + 4) + 5 is same as 3 + (4 + 5).

• Commutative property: The commutative property states that it doesn’t matter what order you use to add the same numbers. Regardless of what number you list first in a set of numbers, they always produce the same sum. So 2+3=5 is the same as 3+2=5.

2. Subtraction

Subtraction is the opposite of addition. You take away a value from another value and end up with the difference. So if 3+4=7, then 7-3=4.

In subtraction, order does matter, so neither the associative property nor the commutative property applies.

3. Multiplication

Multiplication as repeated addition with an end result called the product: For example, 3×5 is the same as 5+5+5. They both equal 15.

Multiplication is like addition, in that the order of the values doesn’t matter. So it obeys the commutative property and the associative property.

• Commutative Property: a×b = b×a
• Associative Property: (a×b)×c = a×(b×c)

Another property associated with multiplication is the distributive property. You solve it by distributing the a to b and c, which means that you multiply a and b to get ab and then a and c to get ac, and then you add the results together.

• Distributive Property: a(b+c) = ab + ac

4. Division

Division is the opposite of multiplication. With division, you split one value into smaller values. The end result is called the quotient. So whereas 3×5=15, 15÷3=5, and 15÷5=3.

As in subtraction, order matters in division also, so division doesn’t follow either the commutative or associative properties. The number at the beginning of any equation using division (15 in the last expression) is called the dividend and the number that goes into the dividend is the divisor (3 or 5 in the last expression).

### Prime Factorisation

This is the rule of Arithmetic. Every positive integer greater than one has a unique prime factorisation. For example, 30 = 2*3*5

### Order of Operations (PEMDAS)

Basic arithmetic requires that you perform the operations in a certain order from left to right. If you have an expression that contains addition, subtraction, multiplication, division, exponents and roots, and parentheses, the acronym PEMDAS can help you remember to perform operations in the following order:

1. Parentheses
2. Exponents and roots
3. Multiplication and Division